Mass transfer models surface-renewal theory

The fundamental principles of the gas-to-liquid mass transfer were concisely presented. The basic mass transfer mechanisms described in the three major mass transfer models the film theory, the penetration theory, and the surface renewal theory are of help in explaining the mass transport process between the gas phase and the liquid phase. Using these theories, the controlling factors of the mass transfer process can be identified and manipulated to improve the performance of the unit operations utilizing the gas-to-liquid mass transfer process. The relevant unit operations, namely gas absorption column, three-phase fluidized bed reactor, airlift reactor, liquid-gas bubble reactor, and trickled bed reactor were reviewed in this entry. 

Other models such as the penetration model (surface renewal theory) developed by Higbie and Danckwerts (Westerterp, van Swaaij, and Beenackers, 1998) consider the mass transfer process to be essentially non-stationary, and the surface is assumed to consist of elements of different age at the surface, returning into the bulk phase while new elements originating from the bulk phase take their place. Results of calculations by the two-film theory and the surface renewal theory are similar. Thus, only the two-film theory, which is easier to understand and which is therefore used most, is considered here. 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

When necessary, results of surface-renewal theories will be presented simultaneously, as these models may be applied directly to determine interfacial area and mass-transfer coefficients in the laboratory apparatus considered in Sections III,B and V. 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

An apparent weakness of the film model is that it suggests that the mass transfer coefficient is directly proportional to the diffusion coefficient raised to the first power. This result is in conflict with most experimental data, as well as with more elaborate models of mass transfer [surface renewal theory considered in the next chapter, e.g., or boundary layer theory (Bird et al., I960)]. However, if we substitute the film theory expression for the mass transfer coefficient (Eq. 8.2.12) into Eq. 8.8.1 for the Sherwood number we find... 

The K j may be estimated using an empirical correlation or alternative physical model (e.g., surface renewal theory) with the Maxwell-Stefan diffusivity of the appropriate i-j pair D-j replacing the binary Fick D. Since most published correlations were developed with data obtained with nearly ideal or dilute systems where F is approximately unity, we expect this separation of diffusive and thermodynamic contributions to k to work quite well. We may formally define the Maxwell-Stefan mass transfer coefficient k - as (Krishna, 1979a)... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

Huang and Kuo also solved two equations for a rapid first-order reversible reaction (i.e., equilibrium in the bulk liquid). The solutions are extremely lengthy and will not be given here. From a comparison of the film, surface renewal, and intermediate film-penetration theories it was found that for irreversible and reversible reactions with equal diffusivities of reactant and product, the enhancement factor was insensitive to the mass transfer model. For reversible reactions with product diffusivity smaller than that of the reactant, the enhancement factor can differ by a factor of two between the extremes of film and surface renewal theory. To conclude, it would seem that the choice of the model matters little for design calculations the predicted differences are negligible with respect to the uncertainties of prediction of some of the model or operation parameters. 

The mass transfer model is based on a physical picture of surface renewal that was developed for describing mass transfer across mobile interfaces. The mass transfer coefficient is then based on the theory for non-steady state diffusion. For relatively short periods of time, the time dependent mass transfer coefficient, according to the penetration theory follows from (see also section 4.62,1)... 

The presentation so far has been carried out in the context of a film theory of mass transfer (see Section 3.1.4) and steady state conditions. There is considerable literature on other models of mass transfer, e.g. surface renewal theory. Further, unsteady state analyses exist for a number of cases. Detailed treatments are available in Danckwerts (1970) and Sherwood et dL (1975). 

Danckwerts surface renewal theory A conceptual mass transfer model used to describe the transfer of a substance from a liquid to a gas. It assumes that an element of the surface interface comprises a mosaic of elements of various ages. Each element has a random chance of being replaced by another element from the bulk of the liquid. A feamre of the model is that a simple mathematical solution is used for complex cases involving chemical reactions. It was formulated by Peter V. Danckwerts (1916-84). 

The value of the surface-renewal theory is that the simple math basic to the penetration theory is extended to a more realistic physical situation. Although the result is less exact than we might wish, the surface-renewal theory does suggest reasonable ways to think about mass transfer in complex situations. Such thoughts can lead to more effective correlations and to better models. 

Kishinev ski/23 has developed a model for mass transfer across an interface in which molecular diffusion is assumed to play no part. In this, fresh material is continuously brought to the interface as a result of turbulence within the fluid and, after exposure to the second phase, the fluid element attains equilibrium with it and then becomes mixed again with the bulk of the phase. The model thus presupposes surface renewal without penetration by diffusion and therefore the effect of diffusivity should not be important. No reliable experimental results are available to test the theory adequately. 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

The results showed that mass transfer through the gas-side boundary layer could be described by the penetration theory (Hygbie 1935) or by the surface renewal model (Danckwerts 1951). It was found that ... 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

Danckwerts8 reasoned that some surfaces may remain in contact longer than other surfaces and generalized the penetration theory by introducing a probability function which expresses the fact that there exists a distribution of contact times lying between zero and infinity. In the resulting expression obtained for Na, the term 2 /D/nt is replaced by yJWs, where s is the fractional renewal rate. Since the unknown t is replaced by the unknown s, the model is not suitable for the prediction of mass transfer coefficients. 

The various forms of the penetration theory can be classified as surface-renewal models, implying either formation of new surfaee at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time or its reciprocal, the average rate of renewal, are functions of the fluid velocity, the fluid properties, the the geometry of the system and can be accurately predicted in only a few special cases. However, even if tj must be determined empirically, the surface-renewal models give a sound basis for correlation of mass-transfer data in many situations, particularly for transfer to drops and bubbles. The similarity between Eqs. (21.44) and (15,20) is an example of the close analogy between heat and mass transfer. It is often reasonable to assume that tj-is the same for both processes and thus to estimate rates of heat transfer from measured mass-transfer rates or vice versa. 

The diffusion theory states that matter is deposited in a continuous way on the surface of a crystal at a rate proportional to the difference in concentration between the bulk and the surface of the crystal. The mathematical analysis is then the same as for other diffusion and mass transfer processes and makes use of the film concept. Sometimes, the film theory is considered to be an oversimplification for crystallization and is replaced by a random surface removal theory (20-23). For both theories the rate of crystal growth (dm/dt) is given by equation XVII, where m, is the mass of solid deposited in time t k, the mass transfer coefficient by diffusion. A, the surface area of the crystal, c, the concentration in the supersaturated solution and Cj, the concentration at the crystal-solution interface (3). For the stagnant film and random surface removal model, equations XVIII and XIX can be used, respectively (3,4) D is the diffusion coefficient, x, the film thickness and f, the fractionai rate of surface renewal. 

Current research falls into one of two schools of thought Calderbank s slip velocity model and Lamont and Scott s eddy turbulence model (Linek et al., 2004 Linek et al., 2005b). Even though both models are penetration-type models, they make very different assumptions. The slip velocity model assumes different behavior for small and large bubbles. It also assumes a significant difference between average velocities for the two phases. The slip velocity and the surface mobility control mass transfer and, in terms of penetration theory, surface renewal. 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

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